SM-based Semantics for Answer Set Programs Containing Conditional Literals and Arithmetic

2025-11-03

AI Summary

Existing semantics for nonmonotonic logic programs with conditional literals and arithmetic rely on infinite propositional translations and grounding, limiting expressiveness and analytical efficiency. Method: This paper introduces, for the first time, a direct semantics that avoids both grounding and infinite-logic translations. Built upon the SM-operator framework, it models conditional literals as nested implications and integrates arithmetic constraints natively. Contribution/Results: The semantics strictly generalizes the classical stable model semantics and is precisely equivalent to established semantics (e.g., Fages’, Ferraris’). It provides a concise, analyzable theoretical foundation for modern Answer Set Programming (ASP) extensions—including conditional literals and arithmetic—enabling more efficient solver design and formal verification.

Technology Category

• Constraint Satisfaction and Optimization: Satisfiability Modulo Theories
• Knowledge Representation and Reasoning: Logic Programming
• Reasoning under Uncertainty: Other Foundations of Reasoning under Uncertainty

Application Category

• Semantics and Knowledge: Data modeling to support human-machine intelligence, including LLMs agents, intelligent system behavior, explanations, and user-friendly interactions
• Search and Retrieval-Augmented AI: Web evaluation methodologies and metrics
• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs

Abstract

Modern answer set programming solvers such as CLINGO support advanced language constructs that improve the expressivity and conciseness of logic programs. Conditional literals are one such construct. They form "subformulas" that behave as nested implications within the bodies of logic rules. Their inclusion brings the form of rules closer to the less restrictive syntax of first-order logic. These qualities make conditional literals useful tools for knowledge representation. In this paper, we propose a semantics for logic programs with conditional literals and arithmetic based on the SM operator. These semantics do not require grounding, unlike the established semantics for such programs that relies on a translation to infinitary propositional logic. The main result of this paper establishes the precise correspondence between the proposed and existing semantics.

Problem

• Defining semantics for logic programs with conditional literals
• Developing non-grounding semantics using SM operator
• Establishing correspondence between proposed and existing semantics

Innovation

• SM operator semantics for conditional literals
• Non-grounding approach to logic programs
• Handles arithmetic and conditional constructs